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Paper 077 Disc
Ecology Letters, (1999) 2 : 0±000
REPORT
Gary R. Huxel
Department of Environmental
Science and Policy and Institute
of Theoretical Dynamics, One
Shields Avenue, University of
California, Davis, CA 95616,
U.S.A.
On the influence of food quality in consumer±
resource interactions
Abstract
While nutrients are an important regulating factor in food webs, no theoretical
studies have examined limits to consumer growth imposed by nutrient concentrations
(i.e. food quality) of their prey. Empirical studies have suggested that nutrients may
play a role in limiting assimilation efficiencies of herbivores. Using a simple food
chain model, I find that prey nutrient concentration does directly influence the
growth rate of consumers and potentially increase the stability of consumer±resource
interactions. This suggests that the strength of trophic cascades and the relative
importance of top±down versus bottom±up control in food webs is significantly
influenced by nutrient availability in food resources of consumers. Additionally, the
results imply that increases in resource input may cause a change in which resource is
limiting and thereby negate any potential ``paradox of enrichment''.
Keywords
Consumer-resource, food quality, multiple limiting resources, paradox of enrichment,
trophic cascade
Ecology Letters (1999) 2 : 0±000
INTRODUCTION
Recent theoretical studies have shown that pelagic food
web structure can have strong effects on community
stability and the strength of trophic cascades (Huxel &
McCann 1998; McCann et al. 1998a, b). A number of
factors have been cited as important in structuring food
webs of pelagic systems, including intraguild predation
and omnivory within the zooplankton guild, food quality
and nutrient stoichiometry, allochthonous inputs, high
diversity, and spatial heterogeneity (Strong 1992; Polis &
Strong 1996; Huxel & McCann 1998; Mackay & Elser
1998; McCann et al. 1998a, b). Whereas other structures
are possible, recent theoretical food web studies have
focused on energy as the currency of choice (Huxel &
McCann 1998; McCann et al. 1998a, b). While these
studies have had some success in explaining food web
dynamics, energetic models have been criticized on the
basis that community dynamics often cannot be predicted
from data on diet or energy flow (Reiners 1986; Polis
1991; White 1993). Recent studies of pelagic systems have
found that organismal stoichiometry of key nutrients
(nitrogen and phosphorus) and the recycling of these
nutrients can constrain food web dynamics (Sterner et al.
1992; Vanni & Layne et al. 1997; Vanni et al. 1997; Elser
& Foster 1998; Elser et al. 1998). Thus in these systems,
energy flow may not be an adequate predictor of food
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web dynamics. Other studies (Sterner & Hesson 1994;
Elser et al. 1996) suggest that herbivores with high
nutrient demands may be limited not by energetic
demands but by the mineral elements of their food.
Similarly, Brett & MuÈller-Navarra (1997) suggest that
highly unsaturated fatty acid content of resource species Ref start
may determine energetic efficiency across the consumer±
resource interaction, thereby influencing the strength of
trophic coupling and trophic cascades in aquatic pelagic
food webs. Thus nutrient stoichiometry or concentrations
may be an important factor missing in energetically based
food chain models. Here I present a food web model in
which population growth of both prey and consumers
depend upon both nutrients and energetics.
Food chain models that incorporate nutrients typically
focus only on the direct influence of nutrients on the
growth of producers and have found that nutrient inputs
tend to destabilize food webs (14). These models have
2
found that the dynamics of the entire system may be
limited by nutrient availability at the producer level (so
that productivity of the system is limited) (DeAngelis
1992; Holt et al. 1994). However, recent empirical studies
have demonstrated that organismal nutrient concentration
can regulate population growth of phytoplankton and
herbivorous zooplankton (Elser et al. 1988, 1996, 1998;
Sterner 1993; Sterner & Hesson 1994; Sterner et al. 1996;
3
Brett & MuÈller-Navarra 1997; Vanni & Layne 1997;
#1999 Blackwell Science Ltd/CNRS
Paper 077 Disc
02 G. R. Huxel
Vanni et al. 1997; Mackay & Elser 1998). Among
herbivorous zooplankton, Daphnia has high requirements
for phosphorus. Over the course of a growing season,
Daphnia can sequester P, thereby reducing the availability
of this resource to the rest of the pelagic system, slowing
the dynamics of the food web (Sterner et al. 1992, 1994;
Elser et al. 1996; Vanni & Layne 1997; Vanni et al. 1997).
McCann et al. (1998b) have found that food web processes
that reduce interaction strengths and/or growth rates tend
to stabilize community dynamics. Thus, one would expect
that incorporating a nutrient-based growth limiting term
for consumers in model food webs also would tend to
stabilize food webs.
To examine the effects of nutrient limitation on
consumer growth, one can begin with a two trophic level
food web model based upon both nutrients and
energetics. The realistic formulation of a food web model
in which consumers are regulated by both nutrients and
energetics represents a strong challenge. How might these
two factors interact to regulate food web dynamics? Can
one assume that the most limiting factor to consumers'
growth regulates the consumer±resource interaction? Or
is one factor just a scalar, with values between 0 and 1, of
maximum potential growth based upon the other factor?
Or alternatively, do nutrients and energetics interact in a
nonlinear multiple resource fashion? Sterner & Hessen
(1994) suggest that when the nutrient to carbon ratio in
food is below the requirement of consumers, the nutrient
becomes limiting in a linear fashion relative to the product
of the concentration of the nutrient times the amount of
food consumed. But at ratios above the requirement, the
consumer will be satiated in terms of nutrients. However,
this functional relationship need not be linear and any
functional response that imposes a limit to growth due to
the ratio of food consumed versus some limiting nutrient
(e.g. minerals, vitamins, fatty acids) will give qualitatively
similar results. Different functional responses will differ
significantly as to where the shift from top±down to
bottom±up control occurs (Fig. 1). Further experimental
work is needed to establish a more general functional form.
For the present model, I hypothesized a saturating type
II functional response of growth versus amount of
nutrient consumed as a general model. This agreed with
Sommer's suggestion that the relationship was analogous
to Droop's model (which is similar to a type II functional
response) (Sommer 1992). Thus, the function that relates
the interaction between nutrient concentration of the prey
(NR) and the nutrient requirements of the consumer (NC)
has a general form of f(NR, NC). In simulations with
nutrient limitation, I use R/[k3(k4 + Ck2)] as the specific
form of this function, where R/k3 equals NR and Ck2
equals NC. In simulations without nutrient limitation,
f(NR, NC) is set to 1. I incorporate nutrient dynamics into
#1999 Blackwell Science Ltd/CNRS
Figure 1 Hypothesized curves for consumer growth versus
nutrient concentration of prey. The switch from top±down
control to bottom±up control may be a gradual or a steep
gradient depending upon the exact relationship between
consumer growth and the nutrient content of their prey (solid
line with arrows). The solid line is a type II functional response
(such as the one used here and similar to the one suggested by
Sommer 1992); the dotted line is a linear response (such as
hypothesized by Sterner & Hessen 1994); and the dashed line is a
type III functional response. Max is the maximum growth rate,
i.e. no nutrient limitation. The type II functional response will
more quickly move from bottom±up control at low nutrient
concentration of the prey to top±down control compared with
the other functional forms. Thus using a type II functional form
may make the model results presented herein conservative.
a two trophic level model modified from the Yodzis &
Innes (1992) consumer±resource energetic model. In
order to focus on the consumer±resource interaction,
the general model system does not include a nutrient
recycling loop, instead nutrients enter into the system
(from an allochthonuos source) at a constant rate. This
framework also allowed for examining the impact of
varying the rate of supply of allochthonous nutrients into
the system, which can significantly influence the dynamics
of the system (DeAngelis 1992; Sommer 1992). This
agrees with the finding of Sommer (1992) where increased
rates of nutrient input resulted in higher algae P content
and eventually increased densities of Daphnia. Thus, I
used the following set of three equations:
dN
rRN
ÿ eN
ˆI ÿ
dt
k1 ‡ N
dR k3 rRN
xc yc CR
ˆ
ÿ
dt
k1 ‡ N
R ‡ R0
…1†
dC
R
xc yc CR
ˆ ÿxc C ‡
dt
k3 …k4 ‡ Ck2 † R ‡ R0
where N is the mass of nutrient in grams; R is the biomass
of the prey or resource species in grams; C is the biomass of
the consumer in grams; I is the amount of allochthonous
nutrient input into the system (gN day71); r is the per
Paper 077 Disc
Title 03
capita growth rate of R per day in terms of nutrient mass
per biomass of R (gN gR71 day71); e is the loss of
nutrient in outflow (gN day71), k1 is the half saturation
point for the functional response between the nutrient and
the prey; k2 is the mass of nutrient per unit of consumer
biomass (gN gC71); k3 is the conversion of nutrient mass
to prey biomass (gR gN71); k4 is the half saturation point
of the functional response between the nutrient and the
consumer; xc is the mass-specific metabolic rate of the
consumer measured relative to the production-to-biomass
ratio of the prey density; yc is a measure of the ingestion
rate per unit metabolic rate of the consumer; and RO is the
half saturation point for the functional response between
the prey and consumer.
The consumer dynamics in system (1) result from a loss
of biomass due to metabolic processes (e.g. starvation)
and the consumer-prey type II functional response
multiplied by the nutrient based limiting function. This
function is dependent upon the ratio of the amount of
nutrient in the prey biomass (NR) and the amount of
nutrient in the consumer biomass (NC). This represents a
measure of the potential for consumer growth based upon
the available nutrient (for the consumers this is the
amount of nutrient in the prey). While this function can
theoretically exceed 1, it has a biological limit of 1: when it
exceeds 1, the growth of the consumer then becomes
limited by energetics (its maximal metabolic and consumption rates, xc and yc). Studies of nitrogen and
phosphorus composition have elucidated patterns that
are instructive to understand the link between consumer's
elemental composition, biochemistry, life history, and
growth (Sterner 1993; Elser & Hassett 1994; Sterner et al.
1994; Elser et al. 1996, 1998). For example, Daphnia, a
herbivorous cladoceran, has a relatively high specific P
concentration (and therefore a low N:P ratio) while
copepods have lower specific P concentrations (and
therefore high N:P). These differences between the two
taxa are partially the result of life history characteristics.
Cladocerans experience greater growth rates and short life
spans, while copepods are longer lived and have lower
growth rates. Thus cladocerans, such as Daphnia, require
higher levels of RNA for protein synthesis and therefore
higher phosphorus contents and lower N:P ratios (Sterner
et al. 1994; Elser et al. 1996). Furthermore, copepods
undergo complete metamorphosis while cladocerans do
not. During development from juvenile stages to adulthood, copepods increase in structural proteins as they
grow in size, resulting in increases in the N:P (as
protein:RNA) ratio. This is not restricted to aquatic
systems; as Drosophilia exhibit similar trends in elemental
composition during metamorphosis (Sterner et al. 1994).
However, in model system (1), consumer nutrient
concentrations were assumed constant.
I examine the dynamics in terms of stability by
comparing the results of system (1) with f(NR, NC) set
to R/[k3(k4 + Ck2)], as opposed to setting this to 1. I also
vary the input of allochthonous nutrient (I) to examine
whether these inputs are stabilizing or not. One would
expect that low to moderate levels of allochthonous inputs
should have a stabilizing effect, but higher levels will
result in a collapse of the system.
Simulation results of three levels of input for f(NR, NC)
set to R/[k3(k4 + Ck2)] or to 1 are given in Fig. 2. When
f(NR, NC) is set to 1, at all levels of nutrient input the
consumer becomes extinct. In simulations with f(NR, NC)
following from system (1), both the consumer and the
prey persist for at least 1000 model days when I 4 0. At
no input (I = 0), the time frame over which both species
persisted depended upon the initial value of N. Low levels
of allochthonous nutrients stabilized the system. Thus in
all cases examined here, setting the function f(NR, NC) to
R/[k3(k4 + Ck2)] stabilizes the system by allowing
increased persistence of the consumer and by increasing
prey density. However, there may be levels of input that
allow for persistence when f(NR, NC) is set to 1, but not
when f(NR, NC) is less than 1.
Increased stability occurs because the function has a
biological limit of 1 and usually is much less than this
limit, thus weakening the strength of the consumer±prey
interaction. In consumer-resource phase space, nutrient
limitation [i.e. f(NR, NC) less than 1] pushes the
zooplankton nullcline to the right, thereby increasing
stability. When the consumer growth-prey nutrient
content function equals one, then the system is only
limited by the rate of consumption by the consumer; if the
growth rate of the prey is great enough, then the system
can exhibit large fluctuations in population size and
perhaps crash. Thus, one can think of the system
approaching the ``paradox of enrichment'' at high prey
growth rates when nutrient concentration of the prey is
not limiting for the consumer. If the nutrient concentration is limiting, the population growth of the consumer is
reduced and the system becomes more stable. This may
explain why the paradox of enrichment is readily found in
model systems, in which only a single resource (currency)
can be limiting, but not in natural systems where there are
numerous potentially limiting resources. Furthermore,
because I began with parameter values for which the
system without nutrient dynamics is highly unstable, these
results may be extrapolated to a wide range of parameter
values for which the system is initially more stable (Fig. 3,
Yodzis & Innes 1992).
One can then ask whether models of two currencies can
result in a paradox of enrichment. In this model system, I
found that input or allochthonous nutrients influence the
dynamics in a manner similar to what Huxel & McCann
#1999 Blackwell Science Ltd/CNRS
4
Paper 077 Disc
04 G. R. Huxel
Figure 2 In the model system, I used the following parameter values in the numerical simulations: xC = 0.4; yC = 2.009; RO = 0.16129;
e = 0.1; k1 = 0.05; k2 = 0.10; k3 = 25.0; k4 = 10.0; r = 0.10 (2,3,18). Three different values for I (0.0, A and D; 0.1, B and E; 1.0, C and
F) were used to represent no input, low input (oligotrophic conditions), and high input (eutrophic conditions). Numerical simulations
are run for the course of a 100-day growing season following a water column turnover that creates an initial pool of nutrients. This time
frame allows the effects of nutrient sequestration to become important. The system continued to run for 1000 model days to determine
whether the system reaches equilibrium when persistence occurred (B). In numerical simulations, the functional form f(NR, NC) was set
to either R/k3(k4 + Ck2) (A±C) or to 1 (D±F). The solid line presents nutrient levels (N), the dashed line represents the resource species
(R), and the dotted line represents the consumer (C).
(1998) found, in that low levels of input stabilize while
higher levels destabilize food web dynamics. Thus at high
nutrient input, a paradox of enrichment occurs suggesting
that in pelagic systems, in which the paradox of
enrichment does not seem to occur, multiple resources
(42) can be limiting (Tilman 1977, 1982; Leibold 1997).
However, this result may not hold in systems where
nutrient concentrations vary greatly across species, and
lower food quality species increase in numbers due to
lower attack rates compared with higher food quality
#1999 Blackwell Science Ltd/CNRS
species (Hessen & Nilssen 1986). Thus, while I set the
nutrient concentration of the prey constant, this probably
does not hold in natural systems where heavy predation
on high quality prey can result in favourable conditions
for low quality prey.
These results have several important predictions. First,
consumer nutrient requirements may limit their growth,
which results in increased food chain stability. Empirical
studies indicate that herbivores exhibit lower assimilation
efficiencies than carnivores due to the large differences in
5
Paper 077 Disc
Title 05
Figure 3 This figure illustrates how nutrient limitation f(NR, NC)
5 1 can move the consumer nullcline to the right of the hump
(peak) of the resource nullcline, thereby moving the system to a
region of stability from a region of instability (DeAngelis 1992).
The unstable situation occurs in the current model when f(NR,
NC) = 1 (see Fig. 2).
nutrient content of their respective resources (Sterner et
al. 1994; Elser et al. 1996), and that herbivorous
zooplankton in general may be limited by nutritionally
inadequate phytoplankton (Brett & MuÈller-Navarra
1997). One could hypothesize that given very low food
quality (in terms of nutrient contents) of producers,
herbivores in these systems exhibit prey selection, intraguild
predation (including cannibalism), or both. Second, any
growth limitation may cause a shift from complete top±
down control via consumption to a more mixed control
wherein bottom±up control via food quality and nutrient
supply can be important. This agrees with the findings of
Brett & Goldman (1996, 1997) who performed metaanalyses on trophic cascade experiments in lake systems,
and found that zooplankton biomass responded to both
predation and nutrient treatments in some systems (see also
McQueen et al. 1986). One would expect that this shift is
gradual and the steepness of the gradient will depend
upon the functional form of the consumer growth±
nutrient concentration relationship (Fig. 1). Furthermore,
the shift from top±down control to bottom±up control
will weaken trophic cascades (Vanni 1996; Strong 1992;
Brett & Goldman 1996, 1997; Brett & MuÈller-Navarra 1997).
Third, the amount of initial input into the system can
significantly influence stability of the system, receiving
little allochthonous inputs. While nutrient recycling was
not included in model system (1), herbivores may provide
a benefit to primary producers by accelerating the rate of
recycling of limiting resources (DeAngelis 1992; Holt et
al. 1994; Vanni 1996). Fourth, one can reason that
nutrient additions (and similarly allochthonous inputs)
that increase nutrient levels above the level of nutrient
limitation, will produce strong trophic cascades (e.g.
increased consumer densities in two trophic level
systems), but only up to a utilization saturation point.
For example, Daphnia may experience growth limitations
due to phosphorus only when its prey has carbon:phosphorus ratios of greater than 300:1 (Sterner et al. 1994).
Given these potential outcomes, consumer and resource
community composition should be greatly influenced by
elemental stoichiometry and nutritional content in general, which will have important consequences on the
strength of trophic cascades (Elser et al. 1988, 1996, 1998;
Sterner 1993; Sterner et al. 1994; Vanni 1996; Brett &
MuÈller-Navarra 1997; Hassett et al. 1997). I suggest that
more experiments need to approximate and parameterize
the functional form of consumer±resource interactions as
influenced by resource nutritional content. Most significantly, the parameters of the functional form, and perhaps
the form itself, may differ among species and among
nutrients (DeMott 1999). This will require experiments
and hypotheses that incorporate consumer growth±
nutrient concentration relationships to delineate conditions for strong trophic cascades and cautions against
making ecosystem level predictions without accounting
for differences among species.
6
ACKNOWLEDGEMENTS
I thank Kevin McCann, Don DeAngelis, Gary Polis,
Carole Hom, Alan Hastings, Paul Stapp, James Umbanhowar, and two anonymous reviewers for discussions and
comments concerning this work. I also acknowledge
support from a National Science Foundation Research
Training Grant (BIR-960226) to the Institute of Theoretical Dynamics at the University of California, Davis.
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BIOSKETCH
Text??
Editor, J. Grover
Manuscript received 29 January 1999
First decision made 22 March 1999
Manuscript accepted 12 April 1999
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